Abstract convex optimal antiderivatives
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 435-454.

Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.

DOI : https://doi.org/10.1016/j.anihpc.2012.01.004
Classification : 47H04,  47H05,  49N15,  52A01
Mots clés : Abstract convexity, Convex function, Cyclically monotone operator, Fitzpatrick function, Lipschitz extension, Maximal monotone operator, Minimal antiderivative, Subdifferential operator
@article{AIHPC_2012__29_3_435_0,
author = {Bartz, Sedi and Reich, Simeon},
title = {Abstract convex optimal antiderivatives},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {435--454},
publisher = {Elsevier},
volume = {29},
number = {3},
year = {2012},
doi = {10.1016/j.anihpc.2012.01.004},
zbl = {1259.47064},
mrnumber = {2926243},
language = {en},
url = {http://archive.numdam.org/item/AIHPC_2012__29_3_435_0/}
}
Bartz, Sedi; Reich, Simeon. Abstract convex optimal antiderivatives. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 435-454. doi : 10.1016/j.anihpc.2012.01.004. http://archive.numdam.org/item/AIHPC_2012__29_3_435_0/

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